Essential Concepts

• The cross product ${\bf{u}}\times{\bf{v}}$ of two vectors ${\bf{u}}=\langle{u_1,u_2,u_3}\rangle$ and ${\bf{v}}=\langle{v_1,v_2,v_3}\rangle$ is a vector orthogonal to both ${\bf{u}}$ and ${\bf{v}}$. Its length is given by $\parallel{\bf{u}}\times{\bf{v}}\parallel=\parallel{\bf{u}}\parallel\cdot\parallel{\bf{v}}\parallel\cdot\sin\theta$, where $\theta$ is the angle between ${\bf{u}}$ and ${\bf{v}}$. Its direction is given by the right-hand rule.
• The algebraic formula for calculating the cross product of two vectors, ${\bf{u}}=\langle{u_1,u_2,u_3}\rangle$ and ${\bf{v}}=\langle{v_1,v_2,v_3}\rangle$, is ${\bf{u}}\times{\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\bf{k}}$.
• The cross product satisfies the following properties for vectors ${\bf{u}}$, ${\bf{v}}$, and ${\bf{w}}$, and scalar $c$:
  •  ${\bf{u}}\times{\bf{v}}=-({\bf{v}}\times{\bf{u}})$
  • ${\bf{u}}\times\left({\bf{v}}+{\bf{w}}\right)={\bf{u}}\times{\bf{v}}+{\bf{u}}\times{\bf{w}}$
  • $c({\bf{u}}\times{\bf{v}})=(c{\bf{u}})\times{\bf{v}}={\bf{u}}\times(c{\bf{v}})$
  • ${\bf{u}}\times{0}={0}\times{\bf{u}}=0$
  • ${\bf{v}}\times{\bf{v}}=0$
  • ${\bf{u}}\cdot({\bf{v}}\times{\bf{w}})=({\bf{u}}\times{\bf{v}})\cdot{\bf{w}}$
• The cross product of vectors  ${\bf{u}}=\langle{u_1,u_2,u_3}\rangle$ and ${\bf{v}}=\langle{v_1,v_2,v_3}\rangle$ is the determinant $\left|\begin{array}{ccc} {\bf{i}} & {\bf{j}} & {\bf{k}} \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3}\end{array}\right|$
• If vectors ${\bf{u}}$ and ${\bf{v}}$ form adjacent sides of a parallelogram, then the area of the parallelogram is given by $\parallel{\bf{u}}\times{\bf{v}}\parallel$.
• The triple scalar product of vectors ${\bf{u}}$, ${\bf{v}}$, and ${\bf{w}}$ is ${\bf{u}}\cdot({\bf{v}}\times{\bf{w}})$.
• The volume of a parallelepiped with adjacent edges given by vectors ${\bf{u}}$, ${\bf{v}}$, and ${\bf{w}}$ is $V=|{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})|$
• If the triple scalar product of vectors ${\bf{u}}$, ${\bf{v}}$, and ${\bf{w}}$ is zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.
• The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.
• Torque $\tau$ measures the tendency of a force to produce rotation about an axis of rotation. If force ${\bf{F}}$ is acting at a distance ${\bf{r}}$ from the axis, then torque is equal to the cross product of ${\bf{r}}$ and ${\bf{F}}$: $\tau={\bf{r}}\times{\bf{F}}$.